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Cayley diagram

C2×#1088⋊C2 #88883

order432
isoclinismisoclinic class of #1088⋊C2
automorphismAut(G) = #13164097 (order 1728) · Inn(G) = #1088⋊C2 · Out(G) = D4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×#1088⋊C2
unambiguous semidirect product#1088⋊C22
unambiguous semidirect product(((C3⋊D3)⋊C3)⋊C2)⋊C22

Splittings

FittingC2×H(3,𝔽3) ⋊ D4
SylowH(3,𝔽3) ⋊ C2×D4

Subgroups

subgroupG/H
center, hypercenterC2#1088⋊C2
commutator, cosocle#272C23
FrattiniC3C2×GO+(4,𝔽2)
FittingC2×H(3,𝔽3)D4
socleC2×C3GO+(4,𝔽2)
hypocenterH(3,𝔽3)C2×D4
LCSG ▷ #272 ▷ H(3,𝔽3)
UCSC1 ◁ C2

145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 20 points:

(3,7,10,20,9,4)(5,13,16,8,11,15)(6,14,18,19,17,12);(6,11)(7,13)(8,12)(10,18)(14,19)(15,20)(16,17);(1,2)

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(#1088⋊C2)⋊C4, C4×#1088⋊C2, #11722304, …
maximal normal subgroup of(#1088⋊C2)⋊C4, C4×#1088⋊C2, #11722304, …
maximal quotient of#11722294, #11722295, #11722296, …
automorphism group ofC3×((C3⋊D3)⋊C3)⋊C2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 530);